I recently begin to teach how to write and read mathematical proofs to some close friends. I started showing that, in math, we use logic to show that statements are true, and our first theorem was:
"The sum of two even integers is always a even integer".
The proof goes like this:
Suppose $a$ and $b$ are two even integers. Thus, by definition of even integers, there exists two integers $c$ and $d$ such that
$ a = 2c$ and $ b = 2d$.
it follows that
$ a + b = 2c + 2d = 2(c+d) $ (by distributive property)
Note that $ c + d$ is a integer, because the sum of two integers is also a integers. Hence there exits a integer (namely $c+d$) such that $a+b$ is a twice that number. therefore, by definition of even integer, $a + b$ is a even integer.
And one of them, ask me the following question :
"How did you prove that the sum of two even integers is a even integers if you only show that the sum of a and b (which are letters not numbers) is a even integer?"
And i couldn't answer him.
Can someone with more experience help me with this? how to explain variables and constants in proofs?
Thanks.
You state at the beginning of the proof that $a$ and $b$ are even integers. They are not letters they are numbers axiomatically. You have assumed this truth as the starting point of the proof. Math always begins with unjustified assumptions which we choose to accept but there is no limit to how skeptical you can be. Ultimately what axioms you choose to accept as true is rather arbitrary and subject to scrutiny. For example Euclidean and non-Euclidian geometries each have axioms that directly contradict each other however they are self-contained and can each be studied separately by agreeing to different axioms as a starting point. As long as the axioms are internally consistent we can have a meaningful conversation about the consequences of those assumptions even if we don't believe them.